Thus my matrix for T has these outputs as its columns. Via some elementary row reduction I concluded that the matrix (with semicolons indicating the end of a row) [1 0 0;0 1/2 1/2;-1 1/2 -1/2] is the inverse of T, call it S.

Moreover, it can be checked that T*S = I, which leads me to suspect that I have the write answer. Does this look about right?